Multinomial experiments

A multinomial experiment is an experiment that has the following properties:

The experiment consists of k repeated trials.

Each trial has a discrete number of possible outcomes.

On any given trial, the probability that a particular outcome will occur is constant.

The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

Examples of Multinomial experiments

Suppose we have an urn containing 9 marbles. Two are red, three are green, and four are blue (2+3+4=9). We randomly select 5 marbles from the urn, with replacement. What is the probability (P(A)) of the event A={selecting 2 green marbles and 3 blue marbles}?

To solve this problem, we apply the multinomial formula. We know the following:

Synergies between Binomial and Multinomial processes/probabilities/coefficients

Example

Suppose we study N independent trials with results falling in one of k possible categories labeled . Let pi be the probability of a trial resulting in the ith category, where . Let Ni be the number of trials resulting in the ith category, where .

For instance, suppose we have 9 people arriving at a meeting according to the following information:

SOCR Multinomial Examples

Suppose we row 10 loaded hexagonal (6-face) dice 8 times and we are interested in the probability of observing the event A={3 ones, 3 twos, 2 threes, and 2 fours}. Assume the dice are loaded to the small outcomes according to the following probabilities of the 6 outcomes (one is the most likely and six is the least likely outcome).

For instance, running the SOCR Dice Experiment 1,000 times with number of dice n=10, and the loading probabilities listed above, we get an output like the one shown below.

Now, we can actually count how many of these 1,000 trials generated the event A as an outcome. In one such experiment of 1,000 trials, there were 8 outcomes of the type {3 ones, 3 twos, 2 threes and 2 fours}. Therefore, the relative proportion of these outcomes to 1,000 will give us a fairly accurate estimate of the exact probability we computed above

.

Note that that this approximation is close to the exact answer above. By the Law of Large Numbers (LLN), we know that this SOCR empirical approximation to the exact multinomial probability of interest will significantly improve as we increase the number of trials in this experiment to 10,000.